Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

Abstract

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.

Appendix

MatLab routine used to evaluate the surface curvatures.

function [K, H, Ki, Kii] = scurvature (X, Y, Z, gs)

% scurvature − compute gaussian, mean and principal curvatures of a surface

%

% [K, H, Ki, Kii] = scurvature (X, Y, Z, gs), where:

% − X, Y, Z are matrix of points on the surface.

% − gs specifies the spacing between points in every direction. (Default gs = 1)

% − K is the gaussian curvature.

% − H is the mean curvature.

% − Ki and Kii are minimum and maximum curvatures at each point.

%

% First Derivatives

[Xu, Xv] = gradient (X, gs);

[Yu, Yv] = gradient (Y, gs);

[Zu, Zv] = gradient (Z, gs);

% Second Derivatives

[Xuu, Xuv] = gradient (Xu, gs);

[Yuu, Yuv] = gradient (Yu, gs);

[Zuu, Zuv] = gradient (Zu, gs);

[Xuv, Xvv] = gradient (Xv, gs);

[Yuv, Yvv] = gradient (Yv, gs);

[Zuv, Zvv] = gradient (Zv, gs);

% Reshape 2D Arrays into vectors

Xu = Xu(:); Yu = Yu(:); Zu = Zu(:);

Xv = Xv(:); Yv = Yv(:); Zv = Zv(:);

Xuu = Xuu(:); Yuu = Yuu(:); Zuu = Zuu(:);

Xuv = Xuv(:); Yuv = Yuv(:); Zuv = Zuv(:);

Xvv = Xvv(:); Yvv = Yvv(:); Zvv = Zvv(:);

Xu = [Xu Yu Zu];

Xv = [Xv Yv Zv];

Xuu = [Xuu Yuu Zuu];

Xuv = [Xuv Yuv Zuv];

Xvv = [Xvv Yvv Zvv];

% First fundamental form coefficients (g11, g12, g22)

g11 = dot (Xu, Xu, 2);

g12 = dot (Xu, Xv, 2);

g22 = dot (Xv, Xv, 2);

% Normal vector (g3)

m = cross (Xu, Xv, 2);

p = sqrt (dot(m, m, 2));

g3 = m./[p p p];

% Second fundamental Coefficients of the surface (b11, b12, b22)

b11 = dot (Xuu, g3, 2);

b12 = dot (Xuv, g3, 2);

b22 = dot(Xvv, g3, 2);

[s, t] = size (Z);

% Gaussian Curvature (K)

K = (b11 . ∗ b22 − b12 .²)./(g11 . ∗ g22 − g12.²);

K = reshape (K, s, t);

% Mean Curvature (H)

H = (g11 . ∗ b22 + g22 . ∗ b11 − 2. ∗ g12. ∗ b12)./(2 ∗ (g11 . ∗ g22 − g12.²));

H = reshape (H, s, t);

%% Principal Curvatures.

Ki = H + sqrt (H.² − K);

Kii = H − sqrt (H.² − K);

% end scurvature